Thanks for including the good photos, and a ruler!
The typical "rubber ducky" antenna is a simple monopole. The center conductor may be straight, but often it is coiled to reduce the physical length of the antenna while maintaining resonance.
There are also "coaxial dipole" antennas which look somewhat like this. The upper element looks like a simple monopole. The lower (i.e. outer) element is usually solid rather than a coil.
But designing an antenna to work on two bands that are in a 2:1 ratio is a challenge. To meet that, I'm guessing the designer started with the upper element as a quarter-wave at the higher frequency. Then the trick: make the lower (outer) element resonate as a half-wave at the lower frequency -- because the bottom end is grounded, not open. In essence you have an asymmetric coaxial dipole with unequal elements, and one end open, the other shorted. I confess to not having seen this design before.
The whole assembly might have been designed with simulation, but they may have simply experimented with a network analyzer until they got decent VSWR and efficiency in the two bands.